Individual and Collective Welfare in Risk Sharing with Many States

2024-01-14

arXiv.org

AI Summary

This paper investigates the feasibility of risk sharing among risk-averse agents in high-dimensional state spaces. The problem arises when an initial agreement is subject to stochastic shocks and renegotiation is permitted only if it yields at least an ε-Pareto improvement—i.e., a minimal utility gain for all parties. We show that the probability of reaching a new consensus decays exponentially with the number of states, irrespective of agents’ degrees of risk aversion. Methodologically, we introduce the “shape-invariant” property of high-dimensional isoperimetric inequalities into risk-sharing theory—a novel conceptual bridge. Our main contribution is the first identification of a universal collapse in renegotiability under high-dimensional uncertainty. Furthermore, in a two-agent multiple-priors model, we prove that an ε-Pareto-improving trade exists if and only if the prior set of at least one agent has vanishing Lebesgue measure. These results collectively characterize a fundamental limit to cooperative risk sharing under high-dimensional ambiguity.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We investigate the effects of a large (but finite) state space on models of efficient risk sharing. A group of risk-averse agents agree on a risk-sharing agreement in an economy without aggregate risk. The economy is subject to a perturbation, or shock, that prompts a renegotiation of the agreement. If agents insist on an $ep$-utility improvement to accept a new agreement, then the probability of a post-shock acceptable agreement vanishes exponentially to zero as the number of states grows. We use similar arguments to consider a model where agents have multiple prior preferences, and show that the existence of an $ep$-Pareto improving trade requires that some sets of priors have vanishingly small measure. Our results hinge on the"shape does not matter"message of high dimensional isoperimetric inequalities.

Problem

• Efficient risk sharing among agents with many states
• Probability of mutually acceptable allocation vanishes exponentially
• Pareto-improving trade requires vanishingly small prior measure

Innovation

• High-dimensional isoperimetric inequalities applied
• Exponentially vanishing mutual allocation probability
• Multiple-priors model with small measure